Hellinger Square-Root Geometry

The Hellinger distance quantifies the similarity between two probability distributions. For Bernoulli distributions P = (p, 1-p) and Q = (q, 1-q), we can map each distribution to a nonnegative unit vector (√p, √(1-p)) and (√q, √(1-q)) on the positive quadrant of the unit circle. The normalized Hellinger distance represents the Euclidean distance (the chord) between these square-root vectors, scaled by 1/√2.

Explore the geometry

Open the Hellinger Square-Root Geometry experiment to visually and interactively compare distributions.

1. Select Probability P = 0.25 and Probability Q = 0.75. Observe the two vectors on the unit circle arc and the chord connecting them.
2. Change Probability P to 0.5. The chord shortens as the vectors move closer, and the Hellinger distance decreases.
3. Set Probability P = 0 and Probability Q = 1. The distributions are completely disjoint. Their square-root vectors lie on the axes, the Bhattacharyya coefficient is exactly 0, and the normalized Hellinger distance is exactly 1. The angle between them is exactly π/2 radians.